A steady solution for Prandtl's self-similar vortex sheet spirals

Abstract

Prandtl's [L. Prandtl, Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben, in: von Kármán, Levi-Cevita (Eds.), Vorträge aus dem Gebiete der Hydro- und Aerodynamik, Springer, Berlin, 1922] self-similar, semi-infinite, free vortex sheets are characterized by a two-dimensional unsteady flow around an invariable exponential spiral. A similar but steady spiral flow has been published by Schmidt and Sparenberg [G.H. Schmidt, J.A. Sparenberg, On the edge singularity of an actuator disk with large constant normal load, J. Ship Research 21 (1977) 125]. This flow is shown to be, kinematically, the steady solution in Prandtl's class of spirals but with a different dynamic boundary condition since it is not a free vortex sheet but carries a constant load. Due to the kinematic similarity, the analysis of Prandtl's spirals by van Kuik [G.A.M. van Kuik, The flow induced by Prandtl's self-similar vortex sheet spirals at infinite distances from the spiral kernel, Eur. J. Mech. B/Fluids 23 (4) (2004) 607–616] is also valid for the spiral with steady flow. As for Prandtl's spirals, the steady spiral flow requires a description on a multi-branched Riemann surface, by which an interpretation in the single-branched two-dimensional surface is impossible.